Single-Valued and Continuous Functions

Functions may also have ranges of the independent variables within which they are single-valued and other ranges in which they are multi-valued. 

Most functions that have discontinuities, such as the one shown, have them only for certain isolated values of the independent variables, and they are continuous for all other values. As long as we stay within the allowed ranges then, the function is 

Multi-valued and discontinuous functions often present difficulties for differentiation and integration. They are mathematically not well-behaved. There is always great difficulty, for example, in fitting an equation of state in the vicinity of critical points (critical points being both mathematical and physical discontinuities). 

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The mapping w = Log z is a single-valued and continuous function in its domain of definition, which is the set of all nonzero complex numbers its range is the strip -n < Im(w) < 0. Thus the function Log z is analytic. 

The boundary condition associated with this equation is that d(R) be single-valued (and continuous) everywhere in R space, since in (64) ijif ad(r q) is now a single-valued (and continuous) function of q. For comparison purposes, let us consider a hypothetical system of N spin-less particles of charge w and global reduced mass p subject to a mutual magnetic interaction described by a time-independent vector potential. s4(R), in addition to a time-independent static interaction described by a potential V(R). The time-independent Schrodinger equation for the corresponding internal wave function 4 (R) is... 

Let a given direction be a v-axis and a plane vertical to a u-axis be a s-t plane. Then, as shown in figure 2, a layer graph is a planar graph contained in a surface represented by a single-valued and continuous function of co class, such as v = f(s,t). The v-axis is called a normal direction of a layer graph. 

The function m((p) must be single-valued and continuous at all points in space in order for T/ ,(0, single-valued and continuous at some point tpo, then the derivative of Yim(9, delta function at the point [Pg.140]

The physical conditions imposed on the wave function i/) are that it should be single-valued and continuous. Obviously then the physical solutions will have m an integer, since W (0.

mn 0, [Pg.84]

According to the first postulate, the state of a physical system is completely described by a state function fifiq, /) or ket T1), which depends on spatial coordinates q and the time t. This function is sometimes also called a state vector or a wave function. The coordinate vector q has components q, q2, , so that the state function may also be written as q, q2, , t). For a particle or system that moves in only one dimension (say along the x-axis), the vector q has only one component and the state vector XV is a function of x and t Tfix, /). For a particle or system in three dimensions, the components of q are x, y, z and I1 is a function of the position vector r and t Tfi r, /). The state function is single-valued, a continuous function of each of its variables, and square or quadratically integrable. 

When a function and its first derivatives are single valued and continuous, the order of differentiation can be reversed, so that... 

As before, we require that an eigenftmction xp x,y,z) and its derivatives xl) x, y, z) must be finite, single valued and continuous. Such functions are called "well-behaved. ... 

We assume that these functions are single-valued and continuous. Modifications appropriate to motion suffering discontinuity will be given later. We consider a surface a-(t) given by a... 

In quantum mechanics the eigenfunctions which are allowed are always chosen from the class of functions which are single-valued and continuous (except at a finite number of points where the function may become infinite) in the complete range of the variables, and which give a finite result when the squares of their absolute values are integrated over the complete range of the variables. If xf/ is such a func-... 

The product of a function and its complex conjugate is always real and is positive everywhere. Accordingly, the wave function itself may be a real or a complex function. At any point x or at any time t, the wave function may be positive or negative. In order that F(x, t)p represents a unique probability density for every point in space and at all times, the wave function must be continuous, single-valued, and finite. Since F(x, /) satisfies a differential equation that is second-order in x, its first derivative is also continuous. The wave function may be multiplied by a phase factor e , where a is real, without changing its physical significance since... 

In the application of Schrodinger s equation (2.30) to specific physical examples, the requirements that (jc) be continuous, single-valued, and square-integrable restrict the acceptable solutions to an infinite set of specific functions (jc), n = 1, 2, 3,. .., each with a corresponding energy value E . Thus, the energy is quantized, being restricted to certain values. This feature is illustrated in Section 2.5 with the example of a particle in a one-dimensional box. 

In order that the eigenfunctions tp, have physical significance in their application to quantum theory, they are chosen from a special class of functions, namely, those which are continuous, have continuous derivatives, are single-valued, and are square integrable. We refer to functions with these properties as well-behaved functions. Throughout this book we implicitly assume that all functions are well-behaved. 

To be a suitable wave function, Sxiip) must be well-behaved, i.e., it must be continuous, single-valued, and quadratically integrable. Thus, pSu vanishes when p oo because Sxi must vanish sufficiently fast. Since Sxi is finite everywhere, pSxi also vanishes at p = 0. Substitution of equations (6.22) and (6.23) into (6.19) shows that Sxiip) is normalized with a weighting fianction w(p) equal to p ... 

Since a sine function is continuous, single-valued and finite for all real values of its argument, the only restriction on Ex — p2x/2m is that it be positive. The energy spectrum is therefore continuous as for a classical particle and the constant k = p/h, or in terms of the de Broglie wavelength, k = 2tt/ is also known as the wave vector of the particle. 

A regular function is a mathematical function satisfying the three conditions of being (i) single-valued (ii) continuous with its first derivatives and (iii) quadratically integrable, i.e. vanishing at infinity. 

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